On Monday, he alleged that EQT may be pursuing the "value-destructive acquisition" in order to enrich its executives.

"EQT's perverse compensation structure in fact incentivizes management to pursue this suboptimal, dilutive acquisition, no matter the cost to EQT shareholders," Rosenstein wrote in a letter to EQT's board of directors filed with the Securities and Exchange Commission.

Rosenstein says that the company's average oil and gas production over three years has a significant impact on management's long-term incentive pay, which makes up the largest part of its compensation.

Since the growth is not measured on a per share basis, Rosenstein claims management can drive up its payout by acquiring new production volume, even if it means diluting the value of its shares to purchase Rice's wells with stock, which Rosenstein believes is undervalued.

Jana Partners holds a 5.8 percent stake in EQT.

EQT's compensation structure encourages managers to make decisions that will increase total shareholder return, EQT spokesperson Natalie Cox said in a statement. She noted that 98 percent of shareholders who cast votes at EQT's last annual shareholder meeting approved the pay program.

The company believes the Rice purchase will generate at least $2.5 billion — and potentially an additional $7.5 billion — in savings, primarily by combining capital spending programs and reducing administrative costs.

"EQT's Board of Directors and Management team remain fully committed to the Rice transaction, which we believe is in the best interest of all shareholders and will deliver significant long-term shareholder value," Cox told CNBC.

The proposed Rice acquisition would help EQT executives achieve the type of production growth they need to deliver in order to receive their maximum annual payout, Rosenstein claims.

EQT's purchase of Rice would significantly add to its assets in the Marcellus and Utica shale regions, which account for much of the growth in U.S. natural gas production. Demand for natural gas is on the rise as more domestic power plants burn the fuel and a number of liquefied natural gas export terminals are slated to open in the coming years.

Rosenstein believes a spinoff of EQT's midstream business, which owns and operates pipelines, would better benefit shareholders. However, he thinks management is opposed to this option because it would remove a "large, stable, and growing" driver of cash bonuses paid out to management for hitting annual earnings targets.

He acknowledged that total shareholder return plays a part in EQT executive compensation, but says this part of the payout structure is "largely insensitive to actual share price changes."